Disease Mapping withWinBUGS and MLwiN

Andrew B. LawsonDepartment of Epidemiology and Biostatistics

University of South Carolina, USA

William J. BrowneSchool of Mathematical Sciences

University of Nottingham, UK

Carmen L. Vidal RodeiroDepartment of Epidemiology and Biostatistics

University of South Carolina, USA

Innodata047085605X.jpg

Disease Mapping with

WinBUGS and MLwiN

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Disease Mapping withWinBUGS and MLwiN

Andrew B. LawsonDepartment of Epidemiology and Biostatistics

University of South Carolina, USA

William J. BrowneSchool of Mathematical Sciences

University of Nottingham, UK

Carmen L. Vidal RodeiroDepartment of Epidemiology and Biostatistics

University of South Carolina, USA

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Lawson, Andrew (Andrew B.)Disease mapping with WinBUGS and MLwiN / Andrew B. Lawson, William J. Browne,Carmen L. Vidal Rodeiro.

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Contents

Preface ix

Notation xi

0.1 Standard notation for multilevel modelling xi

0.2 Spatial multiple-membership models and the MMMC notation xii

0.3 Standard notation for WinBUGS models xii

1 Disease mapping basics 1

1.1 Disease mapping and map reconstruction 2

1.2 Disease map restoration 3

2 Bayesian hierarchical modelling 17

2.1 Likelihood and posterior distributions 17

2.2 Hierarchical models 18

2.3 Posterior inference 19

2.4 Markov chain Monte Carlo methods 20

2.5 Metropolis and Metropolis–Hastings algorithms 21

2.6 Residuals and goodness of fit 26

3 Multilevel modelling 29

3.1 Continuous response models 30

3.2 Estimation procedures for multilevel models 35

3.3 Poisson response models 38

3.4 Incorporating spatial information 42

3.5 Discussion 43

4 WinBUGS basics 45

4.1 About WinBUGS 45

4.2 Start using WinBUGS 47

4.3 Specification of the model 50

4.4 Model fitting 59

4.5 Scripts 64

4.6 Checking convergence 65

v

4.7 Spatial modelling: GeoBUGS 67

4.8 Conclusions 72

5 MLwiN basics 75

5.1 About MLwiN 75

5.2 Getting started 77

5.3 Fitting statistical models 84

5.4 MCMC estimation in MLwiN 94

5.5 Spatial modelling 104

5.6 Conclusions 113

6 Relative risk estimation 115

6.1 Relative risk estimation using WinBUGS 115

6.2 Spatial prediction 137

6.3 An analysis of the Ohio dataset using MLwiN 139

7 Focused clustering: the analysis of

putative health hazards 155

7.1 Introduction 155

7.2 Study design 156

7.3 Problems of inference 158

7.4 Modelling the hazard exposure risk 160

7.5 Models for count data 164

7.6 Bayesian models 166

7.7 Focused clustering in WinBUGS 167

7.8 Focused clustering in MLwiN 190

8 Ecological analysis 197

8.1 Introduction 197

8.2 Statistical models 198

8.3 WinBUGS analyses of ecological datasets 199

8.4 MLwiN analyses of ecological datasets 219

9 Spatially-correlated survival analysis 235

9.1 Survival analysis in WinBUGS 235

9.2 Survival analysis in MLwiN 237

10 Epilogue 251

Appendix 1: WinBUGS code for focused

clustering models 253

A.1 Falkirk example 253

A.2 Ohio example 256

vi Contents

Appendix 2: S-Plus function for conversion

to GeoBUGS format 263

Bibliography 267

Index 275

Contents vii

Statistics in Practice

Human and Biological Sciences

Brown and Prescott – Applied Mixed Models in Medicine

Ellenberg, Fleming and DeMets – Data Monitoring in Clinical Trials:

A Practical Perspective

Lawson, Browne and Vidal Rodeiro – Disease Mapping with WinBUGS and

MLwiN

Marubini and Valsecchi – Analysing Survival Data from Clinical Trials and

Observation Studies

Parmigiani – Modeling in Medical Decision Making: A Bayesian Approach

Senn – Cross-over Trials in Clinical Research, Second Edition

Senn – Statistical Issues in Drug Development

Whitehead – Design and Analysis of Sequential Clinical Trials, Revised

Second Edition

Whitehead – Meta-analysis of Controlled Clinical Trials

Earth and Environmental Sciences

Buck, Cavanagh and Litton – Bayesian Approach to Intrepreting Archaeological

Data

Glasbey and Horam – Image Analysis for the Biological Sciences

Webster and Oliver – Geostatistics for Environmental Scientists

Industry, Commerce and Finance

Aitken – Statistics and the Evaluation of Evidence for Forensic Scientists

Lehtonen and Pahkinen – Practical Methods for Design and Analysis of Complex

Surveys, Second Edition

Ohser andMücklich – Statistical Analysis of Microstructures in Materials Science

Preface

The analysis of disease maps has seen considerable development over the last

decade. This development has been reflected in a fast-increasing literature and

has been matched by the development of software tools. The intersecting areas

of spatial statistical methods development and geographical information

systems (GIS) have both witnessed this growth. With increasing public health

concerns about environmental risks and even bioterrorism, the need for good

methods for analysing spatial health data is immediate. Two major software

tools have now been developed, which allow the modelling of spatially-refer-

enced small area health data. These tools, MLwiN and WinBUGS, both provide

facilities for sophisticated modelling of realistically complex health data. Win-

BUGS was developed to allow the application of a wide range of hierarchical

Bayesian models, exploiting modern computational advances, in particular

Gibbs sampling. MLwiN was developed to allow the fitting of models to multi-

level data where a natural parameter hierarchy exists. Originally, this was

implemented using iterative likelihood and quasi-likelihood estimation methods.

However, the most recent versions of the package have implemented Bayesian

computational methodology and now have many parallel capabilities. Increas-

ingly both packages are being used by researchers and also now there is a desire

to be able to apply such methodology in practical public health applications. In

response to this need, the authors have attempted to provide an introduction to

the methods and types of applications where such modelling is feasible. We do

not claim to provide a comprehensive text on disease mapping and have

confined our attention to the main application of these methods to counted

data, where numbers of cases are recorded within small areas.

This book is designed to be of interest to final-year undergraduate and

graduate level statistics and biostatistics students but will also be of relevance

to epidemiologists and public health workers both in higher education and

beyond. The book provides in the introductory chapters (Chapters 1–5) general

background to disease mapping, Bayesian hierarchical modelling and multilevel

modelling approaches, and basic introductions to the use of WinBUGS and

MLwiN. The latter part of the book is focused on application areas, and is

divided between relative risk estimation (Chapter 6), focused clustering (Chapter

7), ecological analysis (Chapter 8), and finally spatial survival analysis (Chapter

ix

9). Throughout the book we provide clear descriptions of the model program-

ming execution and analysis of and interpretation of results. We have adopted

the philosophy that we would attempt to demonstrate how MLwiN and Win-

BUGS approach the same data example, but also have included examples where

either one or the other packages have limitations. We cannot necessarily hope

to provide definitive answers to how modelling is to be approached in every

case. However, we would hope that we provide useful pointers to the issues and

potential benefits of the approaches described. As both MLwiN and WinBUGS

are evolving packages, it is to be expected that features described here may vary

in the future. However, we have done our best to describe the current or soon-

to-be current form of the packages which is relevant to the potential audience

for this published work. All the material described here is available in WinBUGS

1.4 (see Section 4.8.2 for download information and website www.mrc-bsu.

cam.ac.uk/bugs), and in MLwiN (see section 5.6.1 and website http://multileve-

l.ioe.ac.uk/index.html for more details). Most datasets used in this book are

available to download (with associated WinBUGS code) from the site http://

www.sph.sc.edu/alawson/.

We would like to acknowledge the help and contribution of a number of

people during the development of this work. First, we would like to acknowledge

the help of the MLwiN project team, in particular Jon Rasbash, Harvey Gold-

stein, Amy Burch, Lisa Brennan, Fiona Steele and Min Yang. We would like to

thank Allan Clark, Robin Puett, Lance Waller, Tom Richards, James Hebert,

Alastair Leyland, Sudipto Banerjee, Robert McKeown, Ken Kleinman, Peter

Rogerson, Dan Wartenburg and Martin Kulldorff for support and encourage-

ment in the project. In addition, we would like to acknowledge the data

availability afforded by the sophisticated online public access GIS layer system

developed by, amongst others, Guang Zhao of the South Carolina Department of

Health and Environmental Control. Finally, the continuing support and encour-

agement of Sian Jones and Rob Calver at Wiley Europe must be acknowledged

and is much appreciated.

Andrew Lawson (Columbia, SC, USA)

William Browne (Nottingham, UK)

Carmen Vidal Rodeiro (Columbia, SC, USA)

March 2003

x Preface

Notation

In complex random effects models there is often a myriad of different ‘standard’

notations to represent a statistical model. This is generally because the models

were first discovered by many different authors at roughly the same time and

each author had their own particular notation and style.

0.1 STANDARD NOTATION FOR MULTILEVEL

MODELLING

In this book, in the multilevel modelling sections, as we will be using the MLwiN

software package, we will use the notation used by this software package.

If we consider a three-level nested Normal model, then the standard multi-

level model will be written as

yijk ¼ X� þ vk þ ujk þ eijk, vk � N(0, �2v ), ujk � N(0,�2u ), eijk � N(0, �2e )

Here the fixed effects are represented by �, X is a design matrix, and the randomeffects at levels 1, 2 and 3 are represented by e, u, and v respectively. Level

1 units are indexed i, level 2 units j and level 3 units k. There is a rather

unfortunate notational clash as in disease mapping e is typically used to repre-

sent the expected counts. However, the level 1 residuals disappear from the

equation in the Poisson response multilevel model which minimizes confusion.

A three-level Poisson response model is typically written in MLwiN as

yijk � Poisson(�ijk),log (�ijk) ¼ log (eijk)þ X� þ vk þ ujk,

vk �N(0, �2v ), ujk � N(0,�2u ):(1)

In standard disease mapping � is often used rather than �, and the eijk is oftenput on the right-hand side of the equation.

xi

0.2 SPATIAL MULTIPLE-MEMBERSHIP MODELS AND

THE MMMC NOTATION

The disadvantage of the standard multilevel notation is that it relies on the

nested structure of the model. Browne et al. (2001) consider more general

random effect structures including crossed random effects and multiple-mem-

bership structures. Rather than give an index for each classification (level in a

nested structure) they instead use mapping functions to define the unit in the

classification that a particular observation belongs. For example let us consider

the three-level Poisson model and assume that the levels are counties within

regions within nations. Then in the notation of Browne et al. (2001) we can

write Equation (1) as follows:

yi � Poisson(�i),log (�i) ¼ log (ei)þ X� þ u(3)nation [i] þ u

(2)region[i],

u(3)nation[i] �N(0, �

2u(3)), u

(2)region[i] � N(0, �

2u(2)):

So here we define all terms with respect to the lowest (observation) level which

is labelled i. The functions nation[i] and region[i] are mapping functions that

return the nation and region respectively that observation i belongs to. As the

random part of the model consists of a set of classifications which need not now

be ordered in terms of nesting (and if the model contained crossed effects could

not) we simply define each set of random effects with the letter u but include a

superscript that gives the classification a number. We start numbering from 2 as

1 is reserved for the observation level.

The spatial multiple-membership models that we will consider later can be

easily written in this notation as follows:

yi � Poisson(�i),log (�i) ¼ log (ei)þ X� þ

X

j2neigh(i)w

(3)i,j u

(3)j þ u

(2)region[i],

u(3)j �N(0, �2u(3)), u

(2)region[i] � N(0, �

2u(2)):

Here we have a set of region effects indexed by 2 and a set of neighbour effects

that are indexed by 3.

0.3 STANDARD NOTATION FOR WinBUGS MODELS

In hierarchical models for disease maps, the notation commonly used is slightly

different from that used in multilevel models. The basic Poisson likelihood model

is defined as

xii Notation

yi � Poisson(ei�i),

where ei is the expected count and �i is the relative risk in the ith small area.Note that in the notation of multilevel models, �i ¼ ei�i. It is also common to useli ¼ ei�i, and this form is used in Chapter 7.

Modelling focuses on �i. Here we assume this notation for all the standardanalysis within WinBUGS. In addition to region specific notation we also

introduce space–time notation with a second subscript denoting the time period:

yik � Poisson(eik�ik):

Here, k denotes the relevant time period and the expected count and relative risk

are allowed to vary over time periods.

When random effects are introduced into models it is usual to denote region-

specific uncorrelated heterogeneity as vi, and correlated heterogeneity for the

same unit as ui. This differs slightly from the convention in multilevel models.

In each section the relevant notation for that section is introduced and it is

hoped that any differences between sections will not create difficulties for the

reader.

Notation xiii

1

Disease Mapping Basics

The representation and analysis of maps of disease incidence data is now

established as a basic tool in the analysis of regional public health. One of the

earliest examples of disease mapping is the map of the addresses of cholera

victims related to the locations of water supplies given by Snow (1854). In that

case, the street addresses of victims were recorded and their proximity to

putative pollution sources (water supply pumps) was assessed.

The subject area of disease mapping has developed considerably in recent

years. This growth in interest has led to a greater use of geographical or spatial

statistical tools in the analysis of data both routinely collected for public health

purposes and in the analysis of data found within ecological studies of disease

relating to explanatory variables. The study of the geographical distribution

of disease can have a variety of uses. The main areas of application can

be conveniently broken down into the following classes: (1) disease mapping,

(2) disease clustering, and (3) ecological analysis. In the first class, usually the

object of the analysis is to provide (estimate) the true relative risk of a disease of

interest across a geographical study area (map): a focus similar to the processing

of pixel images to remove noise. Applications for such methods lie in health

services resource allocation, and in disease atlas construction (see, for example,

Pickle et al., 1999). The second class, that of disease clustering, has particular

importance in public health surveillance, where it may be important to be able

to assess whether a disease map is clustered and where the clusters are located.

This may lead to examination of potential environmental hazards. A particular

special case arises when a known location is thought to be a potential pollution

hazard. The analysis of disease incidence around a putative source of hazard

is a special case of cluster detection called focused clustering. The third class,

that of ecological analysis, is of great relevance within epidemiological research,

as its focus is the analysis of the geographical distribution of disease in

relation to explanatory covariates, usually at an aggregated spatial level.

Many issues relating to disease mapping are also found in this area, in addition

to issues relating specifically to the incorporation of covariates.

1

Disease Mapping with WinBUGS and MLwiN A. Lawson, W. Browne and C. Vidal Rodeiro

# 2003 John Wiley & Sons, Ltd ISBN: 0-470-85604-1 (HB)

In this volume, we focus on the issues of modelling. While the focus here is on

statistical methods and issues in disease mapping, it should be noted that the

results of such statistical procedures are often represented visually in mapped

form. Hence, some consideration must be given to the purely cartographic

issues that affect the representation of geographical information. The method

chosen to represent disease intensity on the map, be it colour scheme or

symbolic representation, can dramatically affect the resulting interpretation of

disease distribution. It is not the purpose of this review to detail such cognitive

aspects of disease mapping, but the reader is directed to some recent discussions

of these issues (MacEachren, 1995; Monmonier, 1996; Pickle and Hermann,

1995; Walter, 1993).

1.1 DISEASE MAPPING AND MAP RECONSTRUCTION

To begin, we consider two situations which commonly arise in studies of the

geographic distribution of disease. These situations are defined by the form of

the mapped data which arises in such studies. First a study area or window

is defined and within this area for a fixed period of time the locations of cases

of a specified disease are recorded. These locations are usually residential

addresses (street address or, at a higher spatial scale, zip code (USA) or post

code unit (UK) ). When such addresses are known it is possible to proceed

by direct analysis of the case locations. This is termed case-event analysis.

Often this analysis requires the use of point process models and associated

methodology. This form of analysis is reviewed in Lawson (2001, Chapters 4

and 5) and elsewhere (see, for example, Elliott et al. (2000, Chapter 6) ).

Due to the requirements of medical confidentiality, it is often not possible

to obtain data at this level of resolution and so resort must be made to the

analysis of counts of cases within small areas within the study window. These

small areas are arbitrary regions usually defined for administrative purposes,

such as census tracts, counties, municipalities, electoral wards or health

district regions. Data of this type consist of counts of cases within tracts and

the analysis of this data is termed tract count analysis. In this volume we focus

exclusively on tract count analysis. An example of the analysis of case-event

data with a Bernouilli model using WinBUGS is given in Congdon (2003,

Chapter 7).

Essentially the count is an aggregation of all the cases within the tract. By

aggregation, the individual case spatial references (locations) are lost and

therefore any georeference of the count is related to the tract ‘location’. Often

this is represented by the tract centroid. In a chosen study window there is

found to be m tracts. Denote the counts of disease within the m tracts as

{yi}, i ¼ 1, . . . , m. Figure 1.1 displays a tract count example.This example is of the 46 counties of South Carolina in which were collected

the congenital abnormality death counts for the year 1990.

2 Disease mapping basics

0

7

1

5

11

5

16

0

17

4

00

1

1

71

3

0

0

8

2

13

7

0

8

0

3

2

41

110

1

2

3

3

8

6

143

11

6

0

1

5

Figure 1.1 South Carolina congenital abnormality deaths 1990 by counties.

1.2 DISEASE MAP RESTORATION

1.2.1 Simple statistical representations

The representation of disease-incidence data can vary from pictorial representa-

tion of counts within tracts, to the mapping of estimates from complex models

purporting to describe the structure of the disease events. In this section, we

describe the range of mapping methods from simple representations to model-

based forms. The geographical incidence of disease has as its fundamental unit

of observation, the address location of cases of disease. The residential address

(or possibly the employment address) of cases of disease contains important

information relating to the type of exposure to environmental risks. Often,

however, the exact address locations of cases are not directly available, and

one must use instead counts of disease in arbitrary administrative regions, such

as census tracts or postal districts.

1.2.1.1 Crude representation of disease distribution

The simplest possible mapping form is the depiction of disease rates at specific

sets of locations. For counts within tracts, this is a pictorial representation of the

number of events in the tracts plotted at a suitable set of locations (e.g., tract

centroids). The locations of case-events within a spatially heterogeneous popu-

lation can display a small amount of information concerning the overall pattern

of disease events within a window. However, any interpretation of the structure

Disease map restoration 3

of these events is severely limited by the lack of information concerning the

spatial distribution of the background population which might be ‘at risk’ from

the disease of concern and which gave rise to the cases of disease. This popula-

tion also has a spatial distribution and failure to take account of this spatial

variation severely limits the ability to interpret the resulting case-event map. In

essence, areas of high density of ‘at risk’ population would tend to yield high

incidence of case-events and so, without taking account of this distribution,

areas of high disease intensity could be spuriously attributed to excess disease

risk.

In the case of counts of cases of disease within tracts, similar considerations

apply when crude count maps are constructed. Here, variation in population

density also affects the spatial incidence of disease. It is also important to consider

how a count of cases could be depicted in amapped representation. Countswithin

tracts are totals of events from the whole tract region. If tracts are irregular, then

a decision must be made to either ‘locate’ the count at some tract location (e.g.

tract centroid, however defined) with suitable symbolization, or to represent the

count as a fill colour or shade over the whole tract (choropleth thematic map). In

the former case, the choice of location will affect interpretation. In the latter case,

symbolization choice (shade and/or colour) could distort interpretation also,

although an attempt to represent the whole tract may be attractive.

In general, methods that attempt to incorporate the effect of background ‘at

risk’ population (termed: at risk background) are to be preferred. These are

discussed in the next section.

1.2.1.2 Standardized mortality/morbidity ratios and standardization

To assess the status of an area with respect to disease incidence, it is convenient

to attempt to first assess what disease incidence should be locally ‘expected’ in

the tract area and then to compare the observed incidence with the ‘expected’

incidence. This approach has been traditionally used for the analysis of counts

within tracts. Traditionally, the ratio of observed to expected counts within

tracts is called a Standardized Mortality/Morbidity Ratio (SMR) and this ratio is

an estimate of relative risk within each tract (i.e., the ratio describes the odds of

being in the disease group rather than the background group). The justification

for the use of SMRs can be supported by the analysis of likelihood models with

multiplicative expected risk (see, for example, Breslow and Day, 1987). In

Section 1.2.3.1, we explore further the connection between likelihood models

and tract-based estimators of risk. Figure 1.2 displays the SMR thematic map for

congenital abnormality deaths within South Carolina, USA, for the year 1990

based on expected rates calculated from the South Carolina 1990–1998 state-

wide rate per 1000 births.

Define yi as the observed count of the case disease in the ith tract, and ei as the

expected count within the same tract. Then the SMR is defined as:

4 Disease mapping basics

Congenital abnormality deathsSMR 1990 using 8 year rate

1.51 to 4.1 (9)1.09 to 1.51 (9)0.78 to 1.09 (9)0.5 to 0.78 (9)0 to 0.5 (10)

Figure 1.2 South Carolina congenital abnormality deaths 1990: SMRs.

b��i ¼ yiei: (1:1)

In this case it must be decided whether to express the b��i as fill patterns in eachregion, or to locate the result at some specified tract location, such as the

centroid. If it is decided that these measures should be regarded as continuous

across regions then some further interpolation of b��i must be made (see, forexample, Breslow and Day, 1987, pp. 198–9).

SMRs are commonly used in disease map presentation, but have many

drawbacks. First, they are based on ratio estimators and hence can yield large

changes in estimate with relatively small changes in expected value. In the

extreme, when a (close to) zero expectation is found the SMR will be very large

for any positive count. Also the zero SMRs do not distinguish variation in

expected counts, and the SMR variance is proportional to 1=ei. The SMR isessentially a saturated estimate of relative risk and hence is not parsimonious.

1.2.2 Informal methods

To circumvent the problems associated with SMRs a variety of methods have

been proposed. Some of these are relatively informal or nonparametric and

others highly parametric. In the rest of this volume we will concentrate on

the model-based relative risk estimation methods. However, it is useful here to

present briefly some notes on alternative methods.

One approach to the improvement of relative risk estimation is to employ

smoothing tools on SMRs to reduce the noise. These tools could be based on

Disease map restoration 5

interpolation methods, or more commonly on nonparametric smoothers such as

kernel regression (Nadaraya-Watson, local linear) (Bowman and Azzalini,

1997), and partition methods (Ferreira et al., 2002). A variety of exploratory

data analysis (EDA) methods have also been advocated (see, for example,

Cressie, 1993). These methods usually require the estimation of a smoothing

constant which describes the overall behaviour of the relative risk surface. Some

local methods are also available. Generalized additive models have also been

proposed and these have the advantage of allowing the incorporation of covari-

ates (see, for example, Kelsall and Diggle, 1998).

1.2.3 Basic models

When more substantive hypotheses and/or greater amounts of prior informa-

tion are available concerning the problem, then it may be advantageous to

consider a model-based approach to disease map construction. Model-based

approaches can also be used in an exploratory setting, and if sufficiently general

models are employed then this can lead to better focusing of subsequent

hypothesis generation. In what follows, we consider first likelihood models for

case event data and then discuss the inclusion of extra information in the form

of random effects.

1.2.3.1 Likelihood models

Usually the basic model for case-event data is derived from the following

assumptions:

(1) Individuals within the study population behave independently with respect

to disease propensity, after allowance is made for observed or unobserved

confounding variables.

(2) The underlying at risk background intensity has a continuous spatial

distribution, within a specified boundary.

(3) The case-events are unique, in that they occur as single spatially separate

events.

Assumption (1) above allows the events to be modelled via a likelihood

approach, which is valid conditional on the outcomes of confounder variables.

Further, assumption (2), if valid, allows the likelihood to be constructed with

a background continuous modulating intensity function representing the ‘at

risk’ background. The uniqueness of case-event locations is a requirement of

point process theory (the property called orderliness: see, for example, Daley and

Vere-Jones, 1988), which allows the application of Poisson-process models in

this analysis. Assumption (1) is generally valid for non-infectious diseases. It

6 Disease mapping basics

may also be valid for infectious diseases if the information about current

infectives were known at given time points. Assumption (2) will be valid at

appropriate scales of analysis. It may not hold when large areas of a study

window include zones of zero population (e.g. harbours/industrial zones). Often

models can be restricted to exclude these areas however. Assumption (3) will

usually hold for relatively rare diseases but may be violated when households

have multiple cases and these occur at coincident locations. This may not be

important at more aggregate scales, but could be important at a fine spatial

scale. Remedies for such non-orderliness are the use of de-clustering algo-

rithms (which perturb the locations by small amounts), or analysis at a higher

aggregation level. Note that it is also possible to use a conventional case-control

approach to this problem (Diggle et al., 2000).

In the case of observed counts of disease within tracts, the Poisson-process

assumptions given above mean that the counts are Poisson distributed with, for

each tract, a different expectation. Often at this point a simplifying assumption is

made where the ith tract count expectation is regarded as being a function of a

parameter within a model hierarchy, without considering the spatial continuity

of the intensity. This assumption leads to considerable simplifications and the

distribution of the tract counts is often assumed to be

yi � Poisson(ei�i),

where �i is assumed to be a constant relative risk parameter. In this definitionthe expected value of the count is a multiplicative function of the expected

count/rate (ei) and a relative risk. This is the classic model assumed in many

disease mapping studies. The log-likelihood associated with this model is, bar a

constant, given by:

l ¼Xmi¼1

yi ln (ei�i)�Xmi¼1

ei�i:

Note that by differentiation the saturated maximum likelihood estimator of �i isjust yi=ei, the SMR.

This model makes a number of assumptions. First it is assumed that any

excess risk in a tract will be expressed beyond that described by ei. For example,

the expected rate (ei) can be estimated in a variety of ways. Often external

standardization is used, where known supra-regional rates for different age �sex groups are applied to the local population in each tract. The use of external

standardization alone to estimate the expected counts/rates within tracts may

provide a different map from that provided by a combination of external stand-

ardization and measures of tract-specific deprivation (e.g. deprivation indices

(Carstairs, 1981) ). If any confounding variables are available and can be

included within the estimate of the at risk background, then these should be

Disease map restoration 7

considered for inclusion. Examples of confounding variables could be found

from national census data, particularly relating to socioeconomic measures.

These measures are often defined as ‘deprivation’ indicators, or could relate to

lifestyle choices. For example, the local rate of car ownership or percentage

unemployed within a census tract or other small area, could provide a surrogate

measure for increased risk, due to correlations between these variables and poor

housing, smoking lifestyles, and ill-health. Hence, if it is possible to include such

variables, then any resulting map will display a close representation of the ‘true’

underlying risk surface. When it is not possible to include such variables, it is

sometimes possible to adapt a mapping method to include covariates of this type

within regression setting.

1.2.3.2 Fixed effects

Usually the focus of attention when more sophisticated models are applied in

disease mapping is the relative risk. Hence, all the models we will examine in

this volume will be models for the {�i}. One simple model for the relative riskswould be to suppose that there could be a spatial trend or long-range variation

over the study area. To do this we can construct a model which is a function of

the spatial coordinates of the tract centroids: {x1i, x2i} representing eastings and

northings, say. Simple forms of spatial trend can be modelled by using the

centroid coordinates or functions of the coordinates as covariates and assuming

a regression-type model. As the relative risks must be positive it is usual to

model the logarithm of the relative risk as a linear function. Hence, in this case

we could have:

�i ¼ exp{�0 þ �1x1i þ �2x2i}: (1:2)

This model includes a constant rate (exp {�0}) which captures the overall rateacross the whole study region, and two linear parameters: �1,�2. This modeldescribes a planar trend across the study region, and can be easily extended to

include higher-order trend surfaces by adding power functions of the coordinates.

Here we have used centroid locations as covariates, and indeed this model can be

generalized simplywhenyouobserveother covariatesmeasuredwithin the tracts.

For example it may be possible to include deprivation scores for each tract or

census variables such as percentage unemployed or percentage car ownership. In

general, assume that the intercept (constant rate) term is defined for a variable x0iwhich is 1 for each tract. Hence we can specify the model compactly as

�i ¼ exp{xib},

where x is a m� p matrix consisting of p� 1 covariates, b is a p� 1 parametervector and xi denotes the ith observation row of x.

8 Disease mapping basics

This type of fixed effect model can be fitted in conventional statistical pack-

ages which allow Poisson regression or log-linear modelling. The glm function

in R or S-Plus with a log link and log offset of the {ei} can be used, for example.

1.2.3.3 Random effects

In the sections above some simple approaches to mapping counts within tracts

have been described. These methods assume that once all known and observ-

able confounding variables are included then the resulting map will be clean of

all artefacts and hence depicts the true excess risk surface. However, it is often

the case that unobserved effects could be thought to exist within the observed

data and that these effects should also be included within the analysis. These

effects are often termed random effects, and their analysis has provided a large

literature both in statistical methodology and in epidemiological applications

(see, for example, Manton et al., 1981; Tsutakawa, 1988; Breslow and Clayton,

1993; Clayton, 1991; Best and Wakefield, 1999; Lawson, 2001; Richardson,

2003). Within the literature on disease mapping, there has been a considerable

growth in recent years in modelling random effects of various kinds. In the

mapping context, a random effect could take a variety of forms. In its simplest

form, a random effect is an extra quantity of variation (or variance component)

which is estimable within the map and which can be ascribed a defined

probabilistic structure. This component can affect individuals or can be associ-

ated with tracts or covariates. For example, individuals vary in susceptibility to

disease and hence individuals who become cases could have a random com-

ponent relating to different susceptibility. This is sometimes known as frailty.

Another example is the interpolation of a spatial covariable to the locations of

case events or tract centroids. In that case, some error will be included in the

interpolation process, and could be included within the resulting analysis of

case or count events. Also, the locations of case-events might not be precisely

known or subject to some random shift, which may be related to uncertain

residential exposure. Finally, within any predefined spatial unit, such as tracts

or regions, it may be expected that there could be components of variation

attributable to these different spatial units. These components could have

different forms depending on the degree of prior knowledge concerning the

nature of this extra variation. For example, when observed counts, thought to

be governed by a Poisson distribution, display greater variation than expected

(i.e. variance> mean), it is sometimes described as overdispersion. This over-dispersion can occur for various reasons. Often it arises when clustering occurs

in the counts at a particular scale. It can also occur when considerable numbers

of cells have zero counts (sparseness), which can arise when rare diseases are

mapped. In spatial applications, it is important furthermore to distinguish two

basic forms of extra variation. First, as in the aspatial case, a form of independ-

ent and spatially uncorrelated extra variation can be assumed. This is often

Disease map restoration 9

called uncorrelated heterogeneity (see, for example, Besag et al., 1991). Another

form of random effect is that which arises from a model where it is thought that

the spatial unit (such as case-events, tracts or regions) is correlated with neigh-

bouring spatial units. This is often termed correlated heterogeneity. Essentially, this

form of extra variation implies that there exists spatial autocorrelation between

spatial units (see, for example, Cliff and Ord (1981) for an accessible introduction

to spatial autocorrelation). This autocorrelation could arise for a variety of

reasons. First, the disease of concern could be naturally clustered in its spatial

distribution at the scale of observation. Many infectious diseases display such

spatial clustering, and a number of apparently non-infectious diseases also cluster

(see, for example, Cuzick and Hills, 1991; Glick, 1979). Second, autocorrelation

can be induced in spatial disease patterns by the existence of unobserved environ-

mental or frailty effects. Hence the extra variation observed in any application

could arise from confounding variables that have not been included in the

analysis. In disease mapping examples, this could easily arise when simple

mapping methods are used on SMRs with just basic age–sex standardization.

In the discussion above on heterogeneity, it is assumed that a global measure

of heterogeneity applies to a mapped pattern. That is, any extra variation in

the pattern can be captured by including a general heterogeneity term in the

mapping model. However, often spatially-specific heterogeneity may arise

where it is important to consider local effects as well as, or instead of, general

heterogeneity. To differentiate these two approaches, we use the term specific

and nonspecific heterogeneity. Specific heterogeneity implies that spatial loca-

tions are to be modelled locally; for example, clusters of disease are to be

detected on the map. In contrast, ‘nonspecific’ describes a global approach to

such modelling, which does not address the question of the location of effects. In

this definition, it is tacitly assumed that the locations of clusters of disease can be

regarded as random effects themselves. Hence, there are strong parallels be-

tween image processing tasks and the tasks of disease mapping.

Random effects can take a variety of forms and suitable methods must be

employed to provide correctly estimated maps under models including these

effects. In this section, we discuss simple approaches to this problem from a

frequentist, multilevel and Bayesian viewpoint.

A frequentist approach. In what follows, we use the term ‘frequentist’ to

describe methods that seek to estimate parameters within a hierarchical model

structure. The methods do assume that the random effects have mixing (or prior)

distributions. For example a common assumption made when examining tract

counts is that yi � Poisson(ei�i) independently, and that �i � Gamma(�,�). Thislatter distribution is often assumed for the Poisson relative risk parameter and

provides for a measure of overdispersion relative to the Poisson distribution

itself, depending on the �,� values used. The joint distribution is now given bythe product of a Poisson likelihood and a gamma distribution. At this stage a

choice must be made concerning how the random intensities are to be estimated

or otherwise handled. One approach to this problem is to average over the

10 Disease mapping basics

values of �i to yield what is often called the marginal likelihood. Having averagedover this density, it is then possible to apply standard methods such as max-

imum likelihood. This is usually known as marginal maximum likelihood (see,

for example, Bock and Aitkin, 1981; Aitkin, 1996b). In this approach, the

parameters of the gamma distribution are estimated from the integrated likeli-

hood. A further development of this approach is to replace the gamma density

with a finite mixture. This approach is essentially nonparametric and does not

require the complete specification of the parameter distribution (see, for

example, Aitkin, 1996a).

Although the example specified here concerns tract counts, the method

described above can equally be applied to case-event data, by inclusion of a

random component in the intensity specification.

A Bayesian approach. It is natural to consider modelling random effects

within a Bayesian framework. First, random effects naturally have prior distri-

butions and the joint density discussed above is proportional to the posterior

distribution for the parameters of interest. Hence, the development of full Bayes

and empirical Bayes (posterior approximation) methods has progressed natur-

ally in the field of disease mapping. The prior distribution(s) for the (u, say)parameters in the intensity specification ei�i, have hyperparameters (in thePoisson–gamma example above, these were �,�). These hyperparameters canalso have hyperprior distributions. The distributions chosen for these param-

eters depend on the application. In the full Bayesian approach, inference is

based on the posterior distribution of u given the data. However, as in thefrequentist approach above, it is possible to adopt an intermediate approach

where the posterior distribution is approximated in some way, and subsequent

inference may be made via ‘frequentist-style’ estimation of parameters or by

computing the approximated posterior distribution. In the tract-count example,

approximation via intermediate prior-parameter estimation would involve the

estimation of � and �, followed by inference on the estimated posterior distribu-tion (see, for example, Carlin and Louis, 1996, pp. 67–8).

For count data, a number of examples exist where independent Poisson

distributed counts (with constant within-tract rate) are associated with prior

distributions of a variety of complexity. The earliest examples of such a Bayesian

mapping approach can be found in Manton et al. (1981) and Tsutakawa

(1988). Also, Clayton and Kaldor (1987) developed a Bayesian analysis of

a Poisson likelihood model where yi has expectation ei�i, and found thatwith a prior distribution given by �i � Gamma(�,�), the Bayes estimate of �i isthe posterior expectation:

yi þ �ei þ �

: (1:3)

Hence one could map directly these Bayes estimates. Now, the distribution of �iconditional on yi is Gamma(yi þ �, ei þ �) and a Bayesian approach would

Disease map restoration 11

South Carolina empirical Bayes mean RRestimates congenital abnormality deaths 1990

1.51 to 3.96 (8)1.09 to 1.51 (9)0.78 to 1.09 (10)0.51 to 0.78 (8)0.01 to 0.51 (11)

Figure 1.3 Empirical Bayes mean relative risk (RR) estimates.

require summarization of �i from this posterior distribution. In practice, this isoften obtained by generation of realizations from this posterior and then the

summarizations are empirical (e.g.MarkovChainMonteCarlo (MCMC)methods).

Figure 1.3 displays the empirical Bayes estimates under the Poisson–gamma

model with � and � estimated as in Clayton and Kaldor. Note that in contrast tothe SMR map (Figure 1.2), Figure 1.3 presents a smoother relative risk surface.

Other approaches and variants in the analysis of simple mapping models have

been proposed by Tsutakawa (1988), Marshall (1991) and Devine and Louis

(1994). In the next section, more sophisticated models for the prior structure of

the parameters of the map are discussed.

1.2.4 Advanced Bayesian models

Many of the models discussed above can be extended to include the specification

of prior distributions for parameters and hence can be examined via Bayesian

methods. In general, we distinguish here between empirical Bayes methods and

full Bayes methods, on the basis that any method which seeks to approximate

the posterior distribution is regarded as empirical Bayes (Bernardo and Smith,

1994). All other methods are regarded as full Bayes. This latter category

includes maximum a posteriori estimation, estimation of posterior functionals,

as well as posterior sampling.

12 Disease mapping basics

1.2.4.1 Empirical Bayes methods

The methods encompassed under the definition above are wide-ranging, and

here we will only discuss a subset of relevant methods. The first method

considered by the earliest workers was the evaluation of simplified (constrained)

posterior distributions. Manton et al. (1981) used a direct maximization of a

constrained posterior distribution, Tsutakawa (1988) used integral approxima-

tions for posterior expectations, while Marshall (1991) used a method of

moments estimator to derive shrinkage estimates. Devine and Louis (1994)

further extended this method by constraining the mean and variance of the

collection of estimates to equal the posterior first and second moments.

The second type of method which has been considered in the context of

disease mapping is the use of likelihood approximations. Clayton and Kaldor

(1987) first suggested employing a quadratic normal approximation to a Pois-

son likelihood, with gamma prior distribution for the intensity parameter of the

Poisson distribution and a spatial correlation prior. Extensions to this approach

lead to simple generalized least squares (GLS) estimators for a range of likeli-

hoods (Lawson, 1994; 1997).

A third type is the Laplace asymptotic integral approximation, which has

been applied by Breslow and Clayton (1993) to a generalized linear modelling

framework in a disease mapping example. This integral approximation method

allows the estimation of posterior moments and normalizing integrals (see, for

example, Bernardo and Smith, 1994, pp. 340–4). A further, but different,

integral approximation method is where the posterior distribution is integrated

across the parameter space: that is, the nuisance parameters are ‘integrated out’

of the model. In that case the method of nonparametric maximum likelihood

(NPML) can be employed (Bock and Aitkin, 1981; Aitkin, 1996b; Clayton and

Kaldor, 1987). Another possibility is to employ Linear Bayes methods (Marshall,

1991).

1.2.4.2 Full Bayes methods

Full posterior inference for Bayesian models has now become available, largely

because of the increased use of MCMC methods of posterior sampling. The first

full sampler reported for a disease mapping example was a Gibbs sampler

applied to a general model for intrinsic autoregression and uncorrelated hetero-

geneity by Besag et al. (1991). Subsequently, Clayton and Bernardinelli (1992),

Breslow and Clayton (1993) and Bernardinelli et al. (1995) have adapted this

approach to mapping, ecological analysis and space–time problems.

This has been facilitated by the availability of general Gibbs sampling pack-

ages such as BEAM and BUGS (GeoBUGS and WinBUGS). Such Gibbs sampling

methods can be applied to putative source problems as well as mapping/eco-

logical studies. Alternative, and more general, posterior sampling methods, such

Disease map restoration 13

as the Metropolis–Hastings algorithm, are currently not separately available in

a packaged form, although these methods can accommodate considerable

variation in model specification. WinBUGS does provide such estimators when

non-convex posterior distributions are encountered. Metropolis–Hastings algo-

rithms have been applied in comparison to approximate maximum a posteriori

(MAP) estimation by Lawson et al. (1996) and Diggle et al. (1998); hybrid

Gibbs–Metropolis samplers have been applied to space–time problems by Waller

et al. (1997). In addition, diagnostic methods for Bayesian MCMC sample output

have been discussed for disease mapping examples by Zia et al. (1997). Devel-

opments in this area have been reviewed recently (Lawson et al., 1999; Elliott

et al., 2000; Lawson, 2001).

1.2.5 Multilevel modelling approaches

An alternative to the above specification can be considered where a log-linear

form is specified:

�i ¼ exp{�0 þ vi},

where the random term has a zero mean Gaussian distribution, i.e. vi � N(0,�2v )and �2v is the variance of the random effects v.

This model may be rewritten (in terms of counts rather than rates) as

yi � Poisson(�i), log(�i) ¼ log(ei)þ �0 þ vi:

Here �i ¼ ei�i and the log(ei) are treated as known ‘offset’ terms.Generally, multilevel models (see, for example, Goldstein, 1995) are fitted

to data that possess levels of clustering in their structure. In disease mapping

and geographical applications in general such levels would be different levels of

geographical aggregation, for example, census tracts nested within counties

nested within countries. For each level of geography we could then fit normally

distributed random effects so for example if we had data on census tracts nested

within counties we could fit

yij � Poisson(�ij), log(�ij) ¼ log(eij)þ �0 þ vj þ uij,

where both the county and tract random effects have Gaussian distributions, i.e.

vj � N(0, �2v ) and uij � N(0,�2u ).Poisson response multilevel models can be fitted using either frequentist or

Bayesian approaches. Frequentist approaches generally involve some approxi-

mations, for example the software package MLwiN (Rasbash et al., 2000) uses

quasi-likelihood methods that involve Taylor series approximations (Goldstein,

1991; Goldstein and Rasbash, 1996) to transform the problem so that it can be

14 Disease mapping basics